Integrand size = 21, antiderivative size = 308 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {4 b \text {arctanh}(\sin (c+d x))}{a^5 d}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
b^2*(20*a^6-35*a^4*b^2+28*a^2*b^4-8*b^6)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/ 2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d-4*b*arctanh(sin(d*x+c))/a^ 5/d+1/6*(6*a^6-65*a^4*b^2+68*a^2*b^4-24*b^6)*tan(d*x+c)/a^4/(a^2-b^2)^3/d+ 1/3*b^2*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3+1/6*b^2*(9*a^2-4*b^2)* tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/2*b^2*(12*a^4-11*a^2*b^2 +4*b^4)*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
Time = 6.93 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.35 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=-\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{a^5 \left (a^2-b^2\right )^3 \sqrt {-a^2+b^2} d}+\frac {4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}-\frac {4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{a^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{a^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {b^3 \sin (c+d x)}{3 a^2 (a-b) (a+b) d (a+b \cos (c+d x))^3}+\frac {-11 a^2 b^3 \sin (c+d x)+6 b^5 \sin (c+d x)}{6 a^3 (a-b)^2 (a+b)^2 d (a+b \cos (c+d x))^2}+\frac {-47 a^4 b^3 \sin (c+d x)+50 a^2 b^5 \sin (c+d x)-18 b^7 \sin (c+d x)}{6 a^4 (a-b)^3 (a+b)^3 d (a+b \cos (c+d x))} \] Input:
Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]
Output:
-((b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d)) + (4 *b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a^5*d) - (4*b*Log[Cos[(c + d *x)/2] + Sin[(c + d*x)/2]])/(a^5*d) + Sin[(c + d*x)/2]/(a^4*d*(Cos[(c + d* x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/(a^4*d*(Cos[(c + d*x)/2] + S in[(c + d*x)/2])) - (b^3*Sin[c + d*x])/(3*a^2*(a - b)*(a + b)*d*(a + b*Cos [c + d*x])^3) + (-11*a^2*b^3*Sin[c + d*x] + 6*b^5*Sin[c + d*x])/(6*a^3*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])^2) + (-47*a^4*b^3*Sin[c + d*x] + 5 0*a^2*b^5*Sin[c + d*x] - 18*b^7*Sin[c + d*x])/(6*a^4*(a - b)^3*(a + b)^3*d *(a + b*Cos[c + d*x]))
Time = 2.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3281, 3042, 3534, 3042, 3534, 3042, 3534, 27, 3042, 3480, 3042, 3138, 218, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int \frac {\left (3 a^2-3 b \cos (c+d x) a-4 b^2+3 b^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a^2-3 b \sin \left (c+d x+\frac {\pi }{2}\right ) a-4 b^2+3 b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\left (6 a^4-23 b^2 a^2-2 b \left (6 a^2-b^2\right ) \cos (c+d x) a+12 b^4+2 b^2 \left (9 a^2-4 b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {6 a^4-23 b^2 a^2-2 b \left (6 a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+12 b^4+2 b^2 \left (9 a^2-4 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (6 a^6-65 b^2 a^4+68 b^4 a^2-b \left (18 a^4-7 b^2 a^2+4 b^4\right ) \cos (c+d x) a-24 b^6+3 b^2 \left (12 a^4-11 b^2 a^2+4 b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {6 a^6-65 b^2 a^4+68 b^4 a^2-b \left (18 a^4-7 b^2 a^2+4 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-24 b^6+3 b^2 \left (12 a^4-11 b^2 a^2+4 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {3 \left (8 b \left (a^2-b^2\right )^3-a b^2 \left (12 a^4-11 b^2 a^2+4 b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {\left (8 b \left (a^2-b^2\right )^3-a b^2 \left (12 a^4-11 b^2 a^2+4 b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {8 b \left (a^2-b^2\right )^3-a b^2 \left (12 a^4-11 b^2 a^2+4 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 b \left (a^2-b^2\right )^3 \int \sec (c+d x)dx}{a}-\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\frac {b^2 \left (9 a^2-4 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 b \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
Input:
Int[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]
Output:
(b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((b^2*(9*a ^2 - 4*b^2)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3 *b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*C os[c + d*x])) + ((-3*((-2*b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*A rcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + (8*b*(a^2 - b^2)^3*ArcTanh[Sin[c + d*x]])/(a*d)))/a + ((6*a^6 - 6 5*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*Tan[c + d*x])/(a*d))/(a*(a^2 - b^2)))/(2* a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 ))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.80 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}-\frac {1}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}+\frac {2 b^{2} \left (\frac {-\frac {\left (20 a^{4}+5 a^{3} b -18 a^{2} b^{2}-2 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 a^{4}-29 a^{2} b^{2}+9 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 a^{4}-5 a^{3} b -18 a^{2} b^{2}+2 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 a^{6}-35 a^{4} b^{2}+28 b^{4} a^{2}-8 b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}}{d}\) | \(421\) |
| default | \(\frac {-\frac {1}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}-\frac {1}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}+\frac {2 b^{2} \left (\frac {-\frac {\left (20 a^{4}+5 a^{3} b -18 a^{2} b^{2}-2 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 a^{4}-29 a^{2} b^{2}+9 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 a^{4}-5 a^{3} b -18 a^{2} b^{2}+2 a \,b^{3}+6 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 a^{6}-35 a^{4} b^{2}+28 b^{4} a^{2}-8 b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}}{d}\) | \(421\) |
| risch | \(\text {Expression too large to display}\) | \(1314\) |
Input:
int(sec(d*x+c)^2/(a+cos(d*x+c)*b)^4,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/a^4/(tan(1/2*d*x+1/2*c)-1)+4*b/a^5*ln(tan(1/2*d*x+1/2*c)-1)-1/a^4/ (tan(1/2*d*x+1/2*c)+1)-4*b/a^5*ln(tan(1/2*d*x+1/2*c)+1)+2*b^2/a^5*((-1/2*( 20*a^4+5*a^3*b-18*a^2*b^2-2*a*b^3+6*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^ 3)*tan(1/2*d*x+1/2*c)^5-2/3*(30*a^4-29*a^2*b^2+9*b^4)*a*b/(a^2-2*a*b+b^2)/ (a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(20*a^4-5*a^3*b-18*a^2*b^2+2*a*b^ 3+6*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2* d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3+1/2*(20*a^6-35*a^4*b^2+28*a^2 *b^4-8*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan((a-b) *tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (291) = 582\).
Time = 1.57 (sec) , antiderivative size = 2048, normalized size of antiderivative = 6.65 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="fricas")
Output:
[-1/12*(3*((20*a^6*b^5 - 35*a^4*b^7 + 28*a^2*b^9 - 8*b^11)*cos(d*x + c)^4 + 3*(20*a^7*b^4 - 35*a^5*b^6 + 28*a^3*b^8 - 8*a*b^10)*cos(d*x + c)^3 + 3*( 20*a^8*b^3 - 35*a^6*b^5 + 28*a^4*b^7 - 8*a^2*b^9)*cos(d*x + c)^2 + (20*a^9 *b^2 - 35*a^7*b^4 + 28*a^5*b^6 - 8*a^3*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2) *log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^ 2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + 24*((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2* b^10 + b^12)*cos(d*x + c)^4 + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b ^9 + a*b^11)*cos(d*x + c)^3 + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4* b^8 + a^2*b^10)*cos(d*x + c)^2 + (a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b ^7 + a^3*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1) - 24*((a^8*b^4 - 4*a^6*b ^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cos(d*x + c)^4 + 3*(a^9*b^3 - 4*a^7*b^ 5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cos(d*x + c)^3 + 3*(a^10*b^2 - 4*a^8*b ^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*cos(d*x + c)^2 + (a^11*b - 4*a^9*b^ 3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(6*a^12 - 24*a^10*b^2 + 36*a^8*b^4 - 24*a^6*b^6 + 6*a^4*b^8 + (6*a^9*b ^3 - 71*a^7*b^5 + 133*a^5*b^7 - 92*a^3*b^9 + 24*a*b^11)*cos(d*x + c)^3 + 3 *(6*a^10*b^2 - 59*a^8*b^4 + 110*a^6*b^6 - 77*a^4*b^8 + 20*a^2*b^10)*cos(d* x + c)^2 + (18*a^11*b - 132*a^9*b^3 + 239*a^7*b^5 - 169*a^5*b^7 + 44*a^3*b ^9)*cos(d*x + c))*sin(d*x + c))/((a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4...
\[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, dx \] Input:
integrate(sec(d*x+c)**2/(a+b*cos(d*x+c))**4,x)
Output:
Integral(sec(c + d*x)**2/(a + b*cos(c + d*x))**4, x)
Exception generated. \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (291) = 582\).
Time = 0.63 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.91 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="giac")
Output:
-1/3*(3*(20*a^6*b^2 - 35*a^4*b^4 + 28*a^2*b^6 - 8*b^8)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1 /2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^ 6)*sqrt(a^2 - b^2)) + (60*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 105*a^5*b^4*tan (1/2*d*x + 1/2*c)^5 - 24*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 117*a^3*b^6*tan( 1/2*d*x + 1/2*c)^5 - 24*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 42*a*b^8*tan(1/2* d*x + 1/2*c)^5 + 18*b^9*tan(1/2*d*x + 1/2*c)^5 + 120*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 236*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 152*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 - 36*b^9*tan(1/2*d*x + 1/2*c)^3 + 60*a^6*b^3*tan(1/2*d*x + 1/2*c ) + 105*a^5*b^4*tan(1/2*d*x + 1/2*c) - 24*a^4*b^5*tan(1/2*d*x + 1/2*c) - 1 17*a^3*b^6*tan(1/2*d*x + 1/2*c) - 24*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*a*b ^8*tan(1/2*d*x + 1/2*c) + 18*b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^ 2 + a + b)^3) + 12*b*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 - 12*b*log(abs (tan(1/2*d*x + 1/2*c) - 1))/a^5 + 6*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1 /2*c)^2 - 1)*a^4))/d
Time = 52.47 (sec) , antiderivative size = 7490, normalized size of antiderivative = 24.32 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int(1/(cos(c + d*x)^2*(a + b*cos(c + d*x))^4),x)
Output:
(b*atan(((b*((8*tan(c/2 + (d*x)/2)*(128*b^16 - 128*a*b^15 - 768*a^2*b^14 + 768*a^3*b^13 + 1920*a^4*b^12 - 1920*a^5*b^11 - 2600*a^6*b^10 + 2560*a^7*b ^9 + 2025*a^8*b^8 - 1920*a^9*b^7 - 824*a^10*b^6 + 768*a^11*b^5 + 80*a^12*b ^4 - 128*a^13*b^3 + 64*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5 *a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15 *b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (4*b*((16*(8*a^23*b - 8*a^10*b^14 + 4*a^ 11*b^13 + 52*a^12*b^12 - 25*a^13*b^11 - 143*a^14*b^10 + 63*a^15*b^9 + 217* a^16*b^8 - 87*a^17*b^7 - 193*a^18*b^6 + 73*a^19*b^5 + 95*a^20*b^4 - 36*a^2 1*b^3 - 20*a^22*b^2))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a ^20*b^3 - 5*a^21*b^2) - (32*b*tan(c/2 + (d*x)/2)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/(a^5*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5 *a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15 *b^4 - 5*a^16*b^3 - 5*a^17*b^2))))/a^5)*4i)/a^5 + (b*((8*tan(c/2 + (d*x)/2 )*(128*b^16 - 128*a*b^15 - 768*a^2*b^14 + 768*a^3*b^13 + 1920*a^4*b^12 - 1 920*a^5*b^11 - 2600*a^6*b^10 + 2560*a^7*b^9 + 2025*a^8*b^8 - 1920*a^9*b^7 - 824*a^10*b^6 + 768*a^11*b^5 + 80*a^12*b^4 - 128*a^13*b^3 + 64*a^14*b^2)) /(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^...
Time = 0.21 (sec) , antiderivative size = 3632, normalized size of antiderivative = 11.79 \[ \int \frac {\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x)
Output:
(360*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt (a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**7*b**4 - 630*sqrt(a**2 - b* *2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos( c + d*x)*sin(c + d*x)**2*a**5*b**6 + 504*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d* x)**2*a**3*b**8 - 144*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a*b**10 - 120 *sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a** 2 - b**2))*cos(c + d*x)*a**9*b**2 - 150*sqrt(a**2 - b**2)*atan((tan((c + d *x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**7*b**4 + 462*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt (a**2 - b**2))*cos(c + d*x)*a**5*b**6 - 456*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**3*b* *8 + 144*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/ sqrt(a**2 - b**2))*cos(c + d*x)*a*b**10 - 120*sqrt(a**2 - b**2)*atan((tan( (c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a* *6*b**5 + 210*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2 )*b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**4*b**7 - 168*sqrt(a**2 - b**2)* atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*a**2*b**9 + 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan...